2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by fn(x):= e-nx. Does {{n}n>1 converge uniformly on [0, 1]? Does it converge uniformly on (a, 1) with 0 <a<1? Does it converge uniformly on (0, 1)?
Question 1: Let the functions M(n) and f(n) be defined as follows. if n = 0 (1, M(n) = {3}: M(n − 1) – 2n +1, if n > 0 f(n) = n +1 Prove that M(n) = f(n) for all n > 0.
Exercise 17: Let (an) be a sequence. a) Assume an> 0 for all n E N and lim nan =1+0. Show that an diverges. n=1 b) Assume an> 0 for all N EN and lim n'an=1+0. Show that an converges. nal
5. Let be the function defined by f(x) = -1 3 1.5 if r <0 if 0<x<2 if 3 < r <5 Find the Lebesgue integral of f over (-10,10).
1. (10 points) Let f(x) = S1 (+2) - 1, 2<1 1 - 4 sin(), >1 and g(x) = 5-24 (a) Sketch as much off and g as will fit on the graph below. Draw them in two different (dark) colors or one solid/one dashed, and clearly label the graph. -6 -4 -2 2 4 6 8 10X (b) Find the exact value of the limit below, and explain your reasoning in complete sentences. lim (g(x)) =
3.) Let ak E R with ak > 0 for all k E N. Suppose Σ㎞iak converges. Show that Σί1bk (By definition, for a sequence (ck), we say liCkoo if, for all M ER with Hint: Show that there exists (Ni))ไ1 with N > Nj for all j E N, such that bk there exists a sequence (bk)k of real numbers such that lim converges = oo and M >0, there exists N E N such that ck > M...
6. Let si = 4 and sn +1 (sn +-) for n > 0. Prove lim n→oo sn exists and find limn-oo Sn. (Hint: First use induction to show sn 2 2 and the.show (sn) is decreasing)
9. La ste) defined as follows 9. Let f(x) defined as follows: f(x) = 0 if x < -1 = 2(x + 1)/27 if - 1<x<2 = 2/9 if 2 < x < 5 = 0 otherwise. Find F(u) = f(x)dx, where u E R.
Let fi and f2 be functions such that lim e s f1 (2) = + and such that the limit L2 = lim a s f2 (x) exists. Which one of the following is NOT correct? O limas (f1f2)(x) = 0 if L2 = 0. limas (fi + f2)(x) = too if L2 = -0. Olim as (f1f2) (x) = too if 0 <L2 5+co. lim a s (f1f2)(x) = - it L2 = -. Which one of the following...
PLEASE HELP WITH PROOF!! 8. Let an > 0 for all n in 1. Show that if an converges, then Ĉ vanın converges. [Hint: Expand [van - (1/n)]2.) N =